In this talk we will look at a new variant of the polynomial method which was first used to prove that sets avoiding 3-term arithmetic progressions in groups like Z₄ⁿ (Croot, Lev and myself) and Z₃ⁿ (Ellenberg and Gijswijt) are exponentially small (compared to the size of the group).
We will discuss lower and upper bounds for the size of the extremal subsets, including some recent bounds found by Elsholtz and myself. We will also mention some further applications of the method, for instance, the solution of the Erdős-Szemerédi sunflower conjecture.